0
votes
4answers
22 views

Why is my reasoning wrong in determining how many functions there are from set A to set B?

I am trying to count how many functions there are from a set $A$ to a set $B$. The answer to this (and many textbook explanations) are readily available and accessible; I am not looking for the ...
1
vote
2answers
28 views

Proving a bijection(injection and surjection) over a function

I need some help proving bijections: Suppose f is a function from $$ \mathbb R^2 \rightarrow \mathbb R^2$$ Defined by $$f(x,y) = (ax-by,bx+ay)$$ Where a,b are numbers with $$ a^2 + b^2 \neq 0 $$ ...
1
vote
1answer
31 views

Sets Theory Disproof

I have to disprove the statement: For all sets $S$, if $S$ is a subset of the Natural Numbers, then there must exists some $t ∈ S$ such that $|t|\ge1$ Any hints?
-4
votes
1answer
47 views

How to show that $A-(B\cap \overline {C})\subseteq A\cup (B\cap C)$ [on hold]

How do I solve this? Let A, B and C be sets. Show that $A-(B\cap \overline {C})\subseteq A\cup (B\cap C)$ I can't figure it out. Could I use venn or karnaugh?
0
votes
0answers
47 views

Determine whether the set is finite or infinite. If the set is finite, write down explicit list of all the elements. If the set is infinite, say so.

Determine whether the set is finite or infinite. If the set is finite, write down explicit list of all the elements. If the set is infinite, say so and list five laments of the set. $A = \{ n \in ...
0
votes
1answer
20 views

Algebra of sets application

In the problem below algebra of sets is being evaluated, Venn diagrams are allowed. There's a Modern Languages reading exam where 200 students are being evaluated. The exam content is in French, ...
0
votes
1answer
95 views

Proof Question involving Binary Strings & Sets

I'm just wondering about this question I've been working on for my review homework. I tried to solve it on my own and I feel my proof makes decent sense but not the best sense. Please try to give any ...
0
votes
2answers
51 views

Probability/Set theory problem

The problem is: In some country, there are 3 newspapers. 20% of the population read newspaper A, 16% read B, and 14% read C. 8% of the population read both A and B, 5% read A and C, and 4% ...
1
vote
0answers
32 views

Question concerning defining a particular class of functions

I have a multiset of real numbers $X \subseteq \mathbb{R} $ and I want to create a class of injective function to map the elements of $X$ to the unit interval(so basically a normalization). However ...
1
vote
1answer
22 views

Why is the equality right? (Set-Theory)

Let $A, B$ finite sets, and let $f,g\in A\to B$. Also, Let the equivalence class: $$f \sim g \iff \exists h\in Eq(A,A). f=g\circ h $$ Claim: $$f\sim g \iff \forall b\in B. \left| \left\{ a\in A : ...
0
votes
1answer
45 views

What is the cardinality?

Let $A=\left\{1,2,\cdots,10\right\}$ Let $f,g:A\to A$. Consider the equivalence relation $$ fRg \iff \exists h:A\to A. f=h\circ g$$ where $h$ is invertible. Now, let $g(x)=5$: Why is $\left| ...
1
vote
4answers
49 views

Discrete mathematics subsets

Suppose I have two sets A and B: $$ A = \lbrace 2k-1 : k \in \mathbb{Z}\rbrace$$ $$ B = \lbrace 2l+1 : l \in \mathbb{Z}\rbrace$$ I need to prove that A = B. I know that to prove equality between ...
2
votes
1answer
48 views

Proving that a relation is an equivalence relation

I am having difficulties proving the relation IS an equivalence relation. Let $f: X\longrightarrow Y$ be a function from a set $X$ onto a set $Y$. Let $R$ be the subset of $X \times X$ consisting ...
3
votes
4answers
56 views

Show $P\left(A-B\right)=P\left(A\right)-P\left(A \cap B \right)$

I'm trying to show that, given two events $A,B \in \Omega$ ($\Omega$ is a sample space): $$P\left(A-B\right)=P\left(A\right)-P\left(A \cap B \right)$$ I know $A-B = A \cap B^C$, but I don't know how ...
1
vote
3answers
81 views

Number of images from $\mathbb{N}$ to {0, 1}.

Are the number of images from $\mathbb{N}$ to {0, 1} countably infinite or uncountably infinite? I was thinking of counting in base 2 to make a bijection between $\mathbb{N}$ and {0, 1}. So, a ...
2
votes
1answer
29 views

Is this relation symmetric

$R = \{(X, Y) \in \mathscr{P}(A)^2| X \subset Y \text{ and }X \neq Y \}$ I know that $(X,Y) \in R$ holds true since $X \subset Y$. However I'm unsure if $(Y,X) \in R$ since if $Y \subset X$ then ...
1
vote
1answer
60 views

Is the subset relation on the powerset of a set, with qualification, reflexive?

I was wondering if the subset relation is reflexive? $R = \{(X, Y ) \in P(A)^2\mid X\subseteq Y \text{ and } X \neq Y \}$ I assumed they it was reflexive since for all $X \in P(A), X \subseteq X$ is ...
2
votes
2answers
90 views

ZFC and apples described using only fundamental axioms (complete expanded reasoning)

Let's assume that I'm adding two numbers representing my count of objects I perceive (lets say a green and a blue apple that are consider to be of the same class) and I see them as a set of two apples ...
1
vote
2answers
15 views

Mutually disjoint implying complements in set theory

No homework tag because it is just practice for a final, not for marks: $\text{Let $S, T \subseteq U$. If $S \bigcap T= \emptyset$, then $S$ and $T$ :}$ A) are always complements of each other in ...
0
votes
1answer
49 views

Proper Set Theory Transformation

I was wondering if i am using the Inverse Laws Correctly in this transformation: 1. $\mathrm{A}\cup(\mathrm{B}\cap(\mathrm{A}\cup\mathrm{C})\cap(\mathrm{A}\cup\neg\mathrm{C}))$ 2. ...
1
vote
1answer
68 views

Set Theory Laws

I have been working on the Inclusion Exclusion Principal and came across a problem where I am having difficulty identifying the transformation. Given Information: $\mid\mathrm{U}\mid = \mathrm{50}$ ...
2
votes
2answers
80 views

Number of ways to select numbers, each 1 from different lists without repetition

I want the numbers of ways to select numbers each 1 from different lists without allowing repetition. Eg- List 1 : 5, 100, 1 List 2 : 2 List 3 : 5, 100 List 4 : 2, 5, 100 I want to select 1 ...
1
vote
0answers
39 views

How to describe any partition a set

For ignore of a better word, I will use word "partition" try to describe what I mean. How to describe partition(where over lapping subsets are allowed) of a set mathematically? In another word, ...
0
votes
2answers
70 views

Explanation of the formula $f^{-1}(Y)=\{x \in A |f(x) \in Y\}$ for the preimage of a set

So I found a Definition in the book that goes like this to find the pre-image of a set: $$f^{-1}(Y)=\{x \in A |f(x) \in Y\}$$ Example of the theorem being used: Let $A = \{1,2,3,4,5,6\}$ and ...
3
votes
4answers
87 views

Book/Article recommendation

I am a first year Math major in the university, this summer I want to self study and go over some specific subjects. Firstly, can someone can give a suggestion for a detailed book/article about the ...
1
vote
2answers
31 views

If $R$ is a transitive realation, then $R\circ R\subseteq R$

Here's the question I'm struggling with: Let R be a transitive relation on a set A. Prove the R composed with R is a subset of R. I'm kind of lost on how to prove this. I've started with saying: ...
2
votes
1answer
42 views

Prove a statement with elements for Set Theory

I am stuck on this proofing question and I would like some clarification. Q: $A\subseteq B \iff A\cap B^{\prime} = \emptyset$ I already proved that LHS goes to RHS, but I am confused for the other ...
4
votes
1answer
49 views

Proving a Subset Identity

Working on part A of this problem: I worked out the first part like this: 1) If $A$ is a subset of $B$ then $\forall~x~[x\in A \implies x\in B]$ 2) Same goes for $C$ being a subset of $D$ (If ...
0
votes
2answers
35 views

Subsets and Cardinality

I'm confused on if I should count a subset as one element or if I should count all the elements of that subset when computing cardinality. Example: Given the set $A = \{1,2,3,\{4,5,6\}\}$ does $A$ ...
1
vote
1answer
30 views

Find how many People Like dancing Only,People Like Movies

A survey was conducted among 402 persons regarding their interest in movies,dancing and games it was found that (i) 100 People Like games. (ii) 142 People Like movies or dancing but not games. (iii) ...
1
vote
3answers
34 views

Using set theory to count the possible paths on an XY plane

I'm taking an introductory discrete math course, and we're studying set theory. It's going okay, but I read an example problem which gave me some difficulty. I've included a screenshot of the problem. ...
0
votes
2answers
42 views

Discrete Math and Sets and subsets question

Let Universe be {1,2,3,4,5,6} If A = {1,2,3,4} then |A| = 4, and from this we can see that A is an element of U(universe), but can someone explain to me why {A} is NOT an element of U? I'snt the ...
1
vote
0answers
24 views

A set system generated by a closure operator?

Given a ground set $E$, and a matroid closure operator $\tau$ on $\mathcal P(E)$, we can define a set system $(E,F)$ with $$ F := \{X \in \mathcal P(E): \forall x \in X, x \notin \tau(X-\{x\}) \}$$ ...
0
votes
0answers
41 views

How do we prove that, if $\mathcal{P}(A) \sim \mathcal{P}(B)$, then $A \sim B$? [duplicate]

The converse--if $\ A \sim B$ then $ \mathcal{P}(A) \sim \mathcal{P}(B)$--is very easy to prove. I can't see an immediate, simple proof for the converse case. It seems like a potentially good strategy ...
2
votes
1answer
39 views

Cardinality of all inverse functions (bijections) defined on: $\mathbb{R}\rightarrow \mathbb{R}$

What is the cardinality of all inverse functions defined on: $\mathbb{R}\rightarrow \mathbb{R}$? easy to calculate the upper bound which is $2^\aleph$. ($\aleph$ is the cardinality of the ...
0
votes
0answers
30 views

Notations for elementary set theory

I wish to understand the following notations which defined for every cardinals, $a,b$: $P(a,b) = \left| In(A,B) \right|$ $C(a,b) = \left| P_b(A) \right|$ $E(ab) = \left| Eq(A,B) \right|$ ...
0
votes
0answers
43 views

Simple Math Problem on Interval

It's not clear for me. I see this wikipedia page for a difference of half interval on $\mathbb{R}$ and interval on $\mathbb{R}$? For example $$ \{ (-\infty \le x \le a) \, \left|\, a \in \mathbb{R} ...
1
vote
0answers
23 views

element subset for adjacency matrix

I am trying to create an element of a matrix that is a subset of a larger matrix. However, I am told that my subscripts do not match. I wanted other people's opinion as to what I am doing wrong and ...
1
vote
2answers
88 views

Proofs involving sets - True and False?

Can someone please help me with these True and False questions? I've tried them myself, but I'm not very good at discrete math... Thank you in advance! Any set $A$ and $B$ with $B\subseteq A$ and ...
0
votes
1answer
23 views

Proving the following bijection

Let $F = \lbrace S_1, S_2, \dots, S_n \rbrace$, where $S_i \subset \lbrace 1, 2, \dots, 3m\rbrace$ and define a function $f: F \to \mathbb{N}$ by $$ f(S_i) = \sum_{j \in S_i} (n+1)^{3m-j} $$ then ...
0
votes
1answer
41 views

Help on understanding how to express sets and their relations graphically

Let $A=\{0,1\}, B=\{a,b,c\}, R=id_A, S=\{(a,b),(a,c) \}\cup id_B$ Express graphically the following: $(A,R)+(B,S)\\ (B,S)+(A,R)\\ (A,R)\times(B,S)\\ (B,S)\times(A,R)$ I'm not sure how ...
0
votes
1answer
28 views

Need help understanding transitive relations

My discrete math professor gave an example stating that the following relation is transitive, reflexive, symmetric, and antisymmetric. A = {a,b,c,d} R = {(a,a), (b,b), (c,c), (d,d)} I do not ...
1
vote
2answers
53 views

Question about proving subsets.

I need some help understanding the steps to take to prove subsets. Question: For each of the following universal statements regarding any three finite sets $X, Y$, and $Z$, determine whether it is ...
1
vote
2answers
45 views

How to prove or disprove $P(\overline A) = P(U) - P(A)$

Edit: P(U) and P(A) refer to Power Sets. I don't know how to prove, or disprove, $P(\overline A) = P(U) - P(A)$. My initial thoughts is that the statement is true: If I have a set A in universe U, ...
1
vote
1answer
103 views

Prove or find a counterexample: if $A \subseteq B, B \subseteq C, C \subseteq A$, then $A = B = C$

Proof or find a counterexample:For all sets A;B;C if $A \subseteq B$, $B\subseteq C$, and $C\subseteq A$, then $A = B = C$. I tried doing this but not sure whether going in the right way Let $x\in ...
0
votes
1answer
43 views

Prove equality of set equations

I have to prove that $$A\setminus(A\setminus B)=(B\setminus A)\triangle B$$ I am asked to do that by method, where: we assume that some element $u\in A\setminus (A\setminus B) \Rightarrow u\in A\wedge ...
1
vote
1answer
35 views

Discrete math set theory

If a and b are finite sets then, n(A∩B) = n(A)+n(B)-n(A∪B) will this statement be false? and why please explain
1
vote
3answers
73 views

Prove that $A\subseteq B\Longleftrightarrow A\cap B = A$

In set theory logic mathematics. How would i do the proof for: $A\subseteq B\Longleftrightarrow A\cap B = A$
1
vote
1answer
32 views

Number of relations from A to B with specific domain

I have two sets $A = \{1,2,3,4\}$, $B = \{5,6,7,8,9\}$ I need to find the number of relations from $A$ to $B$ which includes $\{1,2,3\}$ in the domain. It says to use the Inclusion–exclusion ...
-1
votes
1answer
46 views

For all sets A, B, and C, if A ⊆ B, then A∩C ⊆ B∩C

Prove each of these statements. Use equation editor for mathematical symbols, formulas, predicates, equations, and so forth. You may use all the proof techniques we've used so far: direct proof, ...